Question: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}8x+8y-12z&=4 \\16x-18y+20z&=-6 \\6x+2y-8z&=14\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Explanation: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}8x+8y-12z&=4 \\16x-18y+20z&=-6 \\6x+2y-8z&=14\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{8}x+{8}y+({-12})z&=4 \\{16}x+({-18})y+{20}z&=-6 \\{6}x+{2}y+({-8})z&=14\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {8} & {8} & {-12} \\ {16} & {-18} & {20} \\ {6} & {2} & {-8} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {8} & {8} & {-12} \\ {16} & {-18} & {20} \\ {6} & {2} & {-8} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 4 \\ -6 \\ 14 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 8 & 8 & -12 \\ -16 & -18 & 20 \\ 6 & 2 & -8 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 4 \\ -6 \\ 14 \end{array} \right]$